Aircraft: DHC8
Phase of flight: Climb
Anomaly: Aircraft Equipment Problem Critical; Deviation / Discrepancy - Procedural Published Material / Policy; Inflight Event / Encounter Weather / Turbulence

We began our day in ZZZ with a flow delay of approximately 20 minutes. Once airborne we encountered moderate turbulence going into ZZZ1. ZZZ1 to ZZZ2 was our next flight. We flew in moderate turbulence with a 30 kt. wind from the west; and a bumpy approach to landing. We were delayed for 2+ hours due to flow on the next flight to ZZZ2. The flight was mostly moderate turbulence enroute and on landing in ZZZ2 had LLWS (Low-Level Wind Shear) with gains + 15 kts. on final with winds gusting 28+ kts. Our fourth leg was a flight from ZZZ1 to ZZZ2 with only 5 passengers and a light load.I - the FO (First Officer) - was the Pilot Flying. As we were climbing out of ZZZ1 we encountered continuous moderate turbulence and entered IMC conditions around 3000 - 4000 ft. As we turned on a crosswind departure just past 4000 ft. a Master Caution fuselage doors light illuminated. We were then given a climb clearance to 6000 ft. We then set the autopilot to level off at 6000 ft. The Captain as the PM (Pilot Monitoring) opened the checklist; [and] we both realized the autopilot was not leveling off. As the Captain said '6000 ft. level off' I disengaged the autopilot and began to level off. Due to the turbulence; it took some time to level off and [to] keep the aircraft from over speeding. We reached around 6400 ft. and then descended to 6000 ft. ATC advised us of the altitude as we were correcting. Once level we continued with the checklist. The checklist concluded that the aircraft was pressurizing correctly indicating the doors were closed even though the aft cargo door was indicating red on the MFD (Multi-function Flight Display). It allowed PIC (Pilot in Command) discretion to land as soon as practical. We accepted a climb to 12000 ft. and continued east trying to get out of the continuous moderate turbulence and reduce our work load. The Captain called Dispatch at that time. After discussing our options we concluded that we should continue to ZZZ3 runway; rather than return to ZZZ1 in a strong headwind; continuous moderate turbulence and LLWS reported on final with winds gusting to 30+ kts. The Captain informed the flight attendants of the situation; and kept them in the loop. Once on the ground in ZZZ3 it was found that the cargo handle on the aft cargo door had popped out.In such a dynamic environment as ZZZ1; especially when being stepped up in IMC conditions while trying to run an emergency checklist; pushing 'ALT' would have been a better way to level off immediately and control the altitude assigned. Departing ZZZ1 to ZZZ3 at about 4000 ft. the Master Caution fuselage doors light illuminated during moderate turbulence. We were then given a climb clearance to 6000 ft. We advised ATC of our door issue and requested no higher. We set the autopilot up for the level off at 6000 ft. I reached for the checklist book and as we were nearing level off I saw that the AP (Autopilot) was not leveling off. I said 6000 level off and PF (Pilot Flying) disengaged AP. Due to the turbulence it took some time to level and [to] keep speed from increasing. We reached 6500 and then descended back to 6000.Once level we began the checklist which concluded that the aircraft was pressurizing indicating that the doors were closed. We accepted a climb to 12000 and continued east in an attempt to escape the continuous moderate turbulence. Once there; I called Dispatch and advised that due to wind shear and turbulence we did not want to return to ZZZ1 or any airport in the cascade mountain or gorge area. The checklist allowed landing as soon as practical so I chose to continue to ZZZ3 where winds were aligned with the runway and more resources were available on the ground. We also had a 50 kt. tailwind which made the time difference negligible versus turning around and being sequenced to another airport. We landed uneventfully. Our day started in ZZZ with a 20 minute flow delay and a moderate turbulence encounter going to ZZZ1. We then went to ZZZ2 where the winds were 30 kts. from the west and had a bumpy approach to 30. We then had a 2 hour flow delay. By the time we left ZZZ2; the surface winds had diminished but on climb out we were quickly back into moderate turbulence. We landed in ZZZ1 with LLWS (Low Level Wind Shear) and gains of +15 on short final. We only had 5 passengers and few bags. Departing ZZZ1 to ZZZ3 for our fourth leg after a hard day the FO (First Officer) was PF (Pilot Flying) and was working to manage the climb with such a light airplane. After turning on the crosswind departure at about 4000 ft. the Master Caution fuselage doors light illuminated during moderate turbulence. We were then given a climb clearance to 6000 ft. We advised ATC of our door issue and requested no higher. We set the autopilot up for the level off at 6000 ft. I reached for the checklist book and as we were nearing level off I saw that the AP was not leveling off. I said 6000 level off and PF disengaged AP and due to the turbulence it took some time to level and keep speed from increasing; we reached 6500 and then descended back to 6000. ATC advised us of the altitude as we were correcting. Once level; we began the checklist which concluded that the aircraft was pressurizing indicating that the doors were closed even though the aft cargo door was indicating red on MFD (Multi-function Flight Display). It allowed PIC (Pilot in Command) discretion to land as soon as practical. We accepted a climb to 12000 and continued east in an attempt to escape the continuous moderate turbulence and reduce our workload. Once there I called Dispatch and advised that due to wind shear and turbulence we did not want to return to ZZZ1 or any airport in the cascade mountain or gorge area. We had seen standing lenticulars southeast of ZZZ2 earlier and yellow radar returns along the gorge. I chose to continue to ZZZ3 where winds were aligned with the runway; the weather was VMC and more resources were available on the ground. We also had a 50 kt. tailwind which made the time difference negligible versus turning around and being sequenced to another airport. We called the flight attendants and the aft position reported a loud bang had been heard but no noises were currently present. I asked them to report any changes. We landed uneventfully and found that the cargo door handle had popped out.With good weather the obvious choice would have been to return to ZZZ1 but considering the existing weather and the workload it imposed I believe the safest and most 'practical' solution was to continue as planned to a more favorable location. DH8-400 flight crew reported receiving a Master Caution fuselage doors light warning in flight.